Evolution by Subtraction. 195 



values some further properties which may reduce the labour 

 of calculation for the higher values. 



Let us begin by working out a sum in the 5th root, 



Ex. Find the 5th root of 1048576, 

 Here m=5. 



>. u n+i ~u n =20n d + l0n = l0n(2n 2 + l) = 10n{ii' 2 + (in 2 + l)}. 

 Hence the Kule :— Cut off the final digit of the given number, 

 and commence the subtraction at the tens' digit. For tho 

 + l)th subtrahend, add to n 2 the next higher number, mul- 

 tiply the sum by n, and add to the product the previous sub- 

 trahend. Eeplace the final digit to the last remainder. 



n 5 =104857(6 

 3 



104854 

 21 



104833 



78 



104755 

 210 



104545 

 465 



104080 

 903 



103177 

 1596 



101581 

 2628 



98953 

 4095 



94858 

 6105 



88753 



8778 



79975 

 12246 



67729 

 16653 



51076 

 22155 



28921 

 28920 



l a =l 



2 2 =4 



3 2 =9 



4 2 =16 



+2 



+5 



10 



17 



3 



9 



19 



33 



— 



X2 



3 



4 





18 



57 



132 





+3 



21 



78 





21 



78 



210 



5 2 =25 



6 2 =36 



7 2 =49 



8 2 =64 



26 



37 



50 



65 



51 



73 



99 



129 



5 



6 



7 



8 



255 



438 



693 



1032 



210 



465 



903 



1596 



465 



903 



1596 



2628 



9 2 =81 



102=100 



11 2 =121 



12 2 =144 



82 



101 



122 



145 



163 



201 



243 



289 



9 



10 



11 



12 



1467 



2010 



2673 



3468 



2628 



4095 



6105 



8778 



4095 



6105 



8778 



12246 



— — 















,3 2 =169 



14 2 =196 



15 2 =225 





170 



197 



226 





339 



393 



451 





13 



14 



15 



















4407 



5502 



6765 





12246 



16653 



22155 





16653 



22155 



28920 





n — 



- 16 



P2 



